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A variational approach to second order mean field games with density constraints : the stationary case.

Mészáros, Alpár Richárd and Silva, Francisco J. (2015) 'A variational approach to second order mean field games with density constraints : the stationary case.', Journal de mathématiques pures et appliquées., 104 (6). pp. 1135-1159.

Abstract

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain . We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order , the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou and Brenier (see [1]), we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order , the previous arguments do not apply and we prove the existence by means of an approximation argument.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1016/j.matpur.2015.07.008
Publisher statement:© 2015 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Date accepted:No date available
Date deposited:28 February 2020
Date of first online publication:01 July 2015
Date first made open access:28 February 2020

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